Author 
Message 
TAGS:

Hide Tags

Senior Manager
Joined: 01 Feb 2005
Posts: 271

If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]
Show Tags
15 Dec 2009, 15:48
17
This post received KUDOS
58
This post was BOOKMARKED
Question Stats:
23% (05:03) correct
77% (02:30) wrong based on 1504 sessions
HideShow timer Statistics
If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative integers, what is the greatest possible value of x – y? (A) 0 (B) 1 (C) 2 (D) 3 (E) 4 This is a challenge problem on MGMAT. I do not have the answer to the question... But on solving the problem I got the answer of 2 (not sure if it is right)
Official Answer and Stats are available only to registered users. Register/ Login.



Math Expert
Joined: 02 Sep 2009
Posts: 39626

Re: nonnegative integers  MGMAT Challenge [#permalink]
Show Tags
15 Dec 2009, 17:04
23
This post received KUDOS
Expert's post
22
This post was BOOKMARKED
axl_oz wrote: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative integers, what is the greatest possible value of x – y? (A) 0 (B) 1 (C) 2 (D) 3 (E) 4
This is a challenge problem on MGMAT. I do not have the answer to the question... But on solving the problem I got the answer of 2 (not sure if it is right) Suppose you got the answer of 2 for the values of \(x\) and \(y\) as 4 and 2. \(2^4+2^2=4^2+2^2\) > \(42=2\) But if we check for \(y=0\), we'll get: \(2^x+2^0=x^2+0^2\) > \(2^x+1=x^2\) > \(2^x=(x1)(x+1)\) > \(x=3\) \(2^3+2^0=9=3^2+0^2\) \(xy=30=3\) 4 can not be the greatest value as when you increase \(x\) so as \(xy\) to be \(4\), \(2^x+2^y\) will always be more than \(x^2+y^2\).
_________________
New to the Math Forum? Please read this: All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Intern
Joined: 18 Dec 2009
Posts: 13

Re: nonnegative integers  MGMAT Challenge [#permalink]
Show Tags
18 Dec 2009, 20:01
good one! i tried backsolving and started with the middle one and then checked one above and one below. got the result.



Manager
Joined: 09 May 2009
Posts: 204

Re: nonnegative integers  MGMAT Challenge [#permalink]
Show Tags
18 Dec 2009, 20:51
it should be 3 , but my way is hit and trial
_________________
GMAT is not a game for losers , and the moment u decide to appear for it u are no more a loser........ITS A BRAIN GAME



Intern
Joined: 12 Oct 2009
Posts: 16

Re: nonnegative integers  MGMAT Challenge [#permalink]
Show Tags
19 Dec 2009, 07:20
Great Problem! Since your trying to find the greatest value of XY, you just have to assume that Y=0, like Bunel said and then use the "hit and trial" approach like xcusem... Said. The algebratic approach is great too, but I know for me personally it opens up the opportunity for me to make silly mistakes. So I try to not use it unless necessary.
Posted from my mobile device



Intern
Joined: 15 Jan 2013
Posts: 39
Concentration: Finance, Operations
GPA: 4

Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]
Show Tags
08 Feb 2013, 09:27
7
This post received KUDOS
3
This post was BOOKMARKED
axl_oz wrote: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative integers, what is the greatest possible value of x – y?
(A) 0 (B) 1 (C) 2 (D) 3 (E) 4
This is a challenge problem on MGMAT. I do not have the answer to the question... But on solving the problem I got the answer of 2 (not sure if it is right) Since we need to maximize the value of x – y, we can do that in two ways...1)make y negative, which is not possible as per the question...2)make y= 0..putting y=0 you will get an equation in x and on hit and trial method u will get the value of x as 3, which will satisfy the equation.... putting x=3 and y=0, we will get the value of x – y as 3.



Intern
Joined: 22 Sep 2012
Posts: 8
Concentration: Entrepreneurship, Strategy
GPA: 3.3
WE: Business Development (Consumer Products)

Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]
Show Tags
20 Mar 2013, 12:16
I have a question for Bunuel: we know that x−y = ((xy)*2)*1/2 = (x*2+y*22xy)*1/2
so substituting the value of x*2 +y*2 as 2*x + 2*y in the above equation
I got x−y = (2*x + 2*y 2xy)*1/2
then, substituting values for x (taking y=0)...the max value for x can be anything more than 0, coz if you take (x=4) then u'l end up with x−y = 4
please can you help me with this problem, where did I go wrong???



Math Expert
Joined: 02 Sep 2009
Posts: 39626

Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]
Show Tags
21 Mar 2013, 03:39



Intern
Joined: 22 Sep 2012
Posts: 8
Concentration: Entrepreneurship, Strategy
GPA: 3.3
WE: Business Development (Consumer Products)

Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]
Show Tags
21 Mar 2013, 07:35
sorry...what i meant to say was, if you took the eq: x−y = (2*x + 2*y 2xy)*1/2
and substitute the values (x = 4 & y = 0), then we'll end up with x−y = (17)*1/2 (which is almost equal to 4)
but before u mentioned the max value of x−y = 3.
did i do something wrong??



Math Expert
Joined: 02 Sep 2009
Posts: 39626

Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]
Show Tags
21 Mar 2013, 08:19



Manager
Joined: 14 Aug 2005
Posts: 84

Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]
Show Tags
21 Mar 2013, 09:34
Brunel and all, Is it a rule to apply one value as zero whenever it is given: 1) Both x and y are nonnegative integers 2) we need to find the max value of xy What if we are asked to find the min ? how do we solve those questions and also, what would be the approach for min and max value of x+y ? Can u guys pls advise?
_________________
One Last Shot



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7440
Location: Pune, India

Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]
Show Tags
21 Mar 2013, 21:56
12
This post received KUDOS
Expert's post
5
This post was BOOKMARKED
surya167 wrote: Brunel and all,
Is it a rule to apply one value as zero whenever it is given:
1) Both x and y are nonnegative integers 2) we need to find the max value of xy
What if we are asked to find the min ? how do we solve those questions and also, what would be the approach for min and max value of x+y ? Can u guys pls advise? Usually, when you are checking for numbers, you do check for 0. It's often a transition point for patterns. Secondly, the question used the term 'nonnegative integers' instead of 'positive integers'  this means 0 would probably have a role to play. There are no such rules but common sense says that we must not ignore 0. Also, this question doesn't really test max min concepts. It is a direct application of your understanding of exponential relations discussed in the post: http://www.veritasprep.com/blog/2013/01 ... cognition/Now, when we look at the equation, 2^x + 2^y = x^2 + y^2, some things come to mind: 1. It is not very easy to find values that satisfy this equation. 2. But there must be some values which satisfy since we are looking for a value of x – y 3. If x = y = 2, the equation is satisfied since all terms become equal and x – y = 0 which is the minimum value of x – y. Usually, the left hand side will be greater than the right hand side (as discussed in the post, 2^n will usually be greater than x^2 except in very few cases). So we must focus on those 'very few cases'. Also, we need to make x and y unequal. We know (from the post) that 2^4 = 4^2 is one solution so we could put x = 4 while keeping y = 2. The equation will be satisfied and x – y = 2 Now, we also know that 2^x < x^2 when x = 3. So is there a solution there as well? The difference between 2^3 and 3^2 is of 1 so can we create a difference of 1 between the other two terms? Sure! If y = 0, then 2^0 = 1 but 0^2 = 0. So another solution is 2^3 + 2^0 = 3^2 + 0^2. Here, x – y = 3 which is the maximum difference. The reason we can be sure that there are no other values is that as you go ahead of 4 on the number line, 2^n will be greater than n^2 (again, discussed in the post). So both left hand side terms will be greater than the right hand side terms i.e. 2^x > x^2 and 2^y > y^2. So, for no other values can we satisfy this equation. Answer (D)
_________________
Karishma Veritas Prep  GMAT Instructor My Blog
Get started with Veritas Prep GMAT On Demand for $199
Veritas Prep Reviews



Manager
Joined: 16 Jan 2011
Posts: 103

Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]
Show Tags
14 Oct 2013, 12:10
2
This post received KUDOS
the expression xy to reach its maximum we need y to be 0. Hence, we need to find X.
therefore, 2^x+2^y=x^2+^2 > 2^x+1=x^2 what means that x is odd. Only 3 satisfies this equation: 2^3+1=3^2. Hence, x must be equal 3



Intern
Joined: 25 Oct 2013
Posts: 21

Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]
Show Tags
06 Dec 2013, 00:37
axl_oz wrote: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative integers, what is the greatest possible value of x – y?
(A) 0 (B) 1 (C) 2 (D) 3 (E) 4
This is a challenge problem on MGMAT. I do not have the answer to the question... But on solving the problem I got the answer of 2 (not sure if it is right) my answer: x – y for x and y 2^x + 2^y = x^2 + y^2 A. x= y x=y=1 2 + 2 = 1 + 1 wrong B. 1 x=2, y=1 4 + 2 = 4 + 1 wrong C. 2 x=3, y=1 8 + 2 = 9 + 1 rightD. 3 x=4, y =1 16 + 2 = 16 + 1 wrong E. 4 x=5, y =1 32 + 2 = 25 + 1 wrong My answer is C.



Math Expert
Joined: 02 Sep 2009
Posts: 39626

Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]
Show Tags
06 Dec 2013, 02:52
misanguyen2010 wrote: axl_oz wrote: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative integers, what is the greatest possible value of x – y?
(A) 0 (B) 1 (C) 2 (D) 3 (E) 4
This is a challenge problem on MGMAT. I do not have the answer to the question... But on solving the problem I got the answer of 2 (not sure if it is right) my answer: x – y for x and y 2^x + 2^y = x^2 + y^2 A. x= y x=y=1 2 + 2 = 1 + 1 wrong B. 1 x=2, y=1 4 + 2 = 4 + 1 wrong C. 2 x=3, y=1 8 + 2 = 9 + 1 rightD. 3 x=4, y =1 16 + 2 = 16 + 1 wrong E. 4 x=5, y =1 32 + 2 = 25 + 1 wrong My answer is C. Please note that the correct answer is D, not C.
_________________
New to the Math Forum? Please read this: All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Intern
Joined: 25 Oct 2013
Posts: 21

Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]
Show Tags
06 Dec 2013, 10:47
1
This post was BOOKMARKED
Bunuel wrote: misanguyen2010 wrote: axl_oz wrote: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative integers, what is the greatest possible value of x – y?
(A) 0 (B) 1 (C) 2 (D) 3 (E) 4
This is a challenge problem on MGMAT. I do not have the answer to the question... But on solving the problem I got the answer of 2 (not sure if it is right) my answer: x – y for x and y 2^x + 2^y = x^2 + y^2 A. x= y x=y=1 2 + 2 = 1 + 1 wrong B. 1 x=2, y=1 4 + 2 = 4 + 1 wrong C. 2 x=3, y=1 8 + 2 = 9 + 1 rightD. 3 x=4, y =1 16 + 2 = 16 + 1 wrong E. 4 x=5, y =1 32 + 2 = 25 + 1 wrong My answer is C. Please note that the correct answer is D, not C. Hi thank you for your reply. I explained what i confused. Of course I read previous answers and all chose D. However, from what i found, i chose C. That s why i posted here. I dont know which is wrong in my answer. Please help!



Math Expert
Joined: 02 Sep 2009
Posts: 39626

Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]
Show Tags
07 Dec 2013, 05:52
misanguyen2010 wrote: Bunuel wrote: misanguyen2010 wrote: my answer: x – y for x and y 2^x + 2^y = x^2 + y^2 A. x= y x=y=1 2 + 2 = 1 + 1 wrong B. 1 x=2, y=1 4 + 2 = 4 + 1 wrong C. 2 x=3, y=1 8 + 2 = 9 + 1 right D. 3 x=4, y =1 16 + 2 = 16 + 1 wrong E. 4 x=5, y =1 32 + 2 = 25 + 1 wrong
My answer is C. Please note that the correct answer is D, not C. Hi thank you for your reply. I explained what i confused. Of course I read previous answers and all chose D. However, from what i found, i chose C. That s why i posted here. I dont know which is wrong in my answer. Please help! To get the greatest value of xy as 3 consider x=3 and y=0. Notice that these values satisfy \(2^x + 2^y = x^2 + y^2\) > \(2^3 + 2^0 =9= 3^2 + 0^2\). Hope it helps.
_________________
New to the Math Forum? Please read this: All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Intern
Joined: 11 Dec 2013
Posts: 19
Location: India
Concentration: Finance, Technology
GMAT Date: 03152015
WE: Operations (Telecommunications)

Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]
Show Tags
12 Dec 2013, 02:02
For x>4,y>4 2^x > x^2  2^y > y^2 Hence maximum we need to check till (x,y) = {0,1,2,3}
By hit and trial .
For x=3,y=1 and x=4,y=2 xy = 2
Hence the answer is 2. Draw the graph for 2^x and x^2. The solution becomes simpler .



Senior Manager
Status: Student
Joined: 26 Aug 2013
Posts: 256
Location: France
Concentration: Finance, General Management
GPA: 3.44

Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]
Show Tags
28 Dec 2013, 04:25
Here is how I done it: 1) If xy needs to be max then Y=0, because Y² is only positive 2) Check the answers, those are only integers, you are therefore looking for an integer 3) You have the equation 2^x +1 = X² 4) Use the different choices and you will see that only 3 matches. Answer D Hope it helps!
_________________
Think outside the box



GMAT Club Legend
Joined: 09 Sep 2013
Posts: 15939

Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]
Show Tags
17 Feb 2015, 07:44
Hello from the GMAT Club BumpBot! Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up  doing my job. I think you may find it valuable (esp those replies with Kudos). Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
GMAT Books  GMAT Club Tests  Best Prices on GMAT Courses  GMAT Mobile App  Math Resources  Verbal Resources




Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte
[#permalink]
17 Feb 2015, 07:44



Go to page
1 2
Next
[ 30 posts ]




